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Stability estimates of inverse random source problems for the wave equations by using correlation-based data

Peijun Li, Ying Liang, Xu Wang

Abstract

This paper focuses on stability estimates of the inverse random source problems for the polyharmonic, electromagnetic, and elastic wave equations. The source is represented as a microlocally isotropic Gaussian random field, which is defined by its covariance operator in the form of a classical pseudo-differential operator. The inverse problem is to determine the strength function of the principal symbol by exploiting the correlation of far-field patterns associated with the stochastic wave equations at a single frequency. For the first time, we show in a unified framework that the optimal Lipschitz-type stability can be attained across all the considered wave equations through the utilization of correlation-based data.

Stability estimates of inverse random source problems for the wave equations by using correlation-based data

Abstract

This paper focuses on stability estimates of the inverse random source problems for the polyharmonic, electromagnetic, and elastic wave equations. The source is represented as a microlocally isotropic Gaussian random field, which is defined by its covariance operator in the form of a classical pseudo-differential operator. The inverse problem is to determine the strength function of the principal symbol by exploiting the correlation of far-field patterns associated with the stochastic wave equations at a single frequency. For the first time, we show in a unified framework that the optimal Lipschitz-type stability can be attained across all the considered wave equations through the utilization of correlation-based data.

Paper Structure

This paper contains 10 sections, 12 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Random scalar field
  4. Random vector field
  5. Polyharmonic waves
  6. The direct problem
  7. The inverse problem
  8. Electromagnetic waves
  9. Elastic waves
  10. Conclusion

Key Result

Lemma 2.3

Let $f$ be a GMIG random field satisfying Assumption assum:1. Then $f\in H^{-\frac{d-m}{2}-\epsilon}(B_1)$ almost surely for any sufficiently small $\epsilon>0$.

Theorems & Definitions (28)

  • Definition 1: cf. LPS08
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 18 more